How to create an orthogonal matrix in Matlab with one fixed column

I want to create a square NxN matrix orthogonal, with the constraint that the first column is a column vector of k*ones(N,1), where k is a constant at choice. Is there any procedure?

I.e.

A= [k * *;k * *;k * *]

is a 3x3 matrix, where the first column is a vector k*ones(3,1), and the other two vectors have to be created in such a way the matrix is orthogonal

Maybe this should be posted on Math.StackEchange where equations can be typed properly if you want a proper theoretical explanation. But if just want the code...

First, if N is the dimension of your matrix, this constrains the value of k to:

k=sqrt(1/N);
A(1,:) = k*ones(1,N);

Then, the second row can be constructed with:

A(2,:) = sqrt(0.5)*[1,-1,zeros(1,N-2)];

This creates a simple vector orthogonal to the first.

The third line can be computed with:

aux = [1,1,-2,zeros(1,N-3)];
A(3,:) = aux/norm(aux);

The fourth:

aux = [1,1,1,-3,zeros(1,N-4)];
A(4,:) = aux/norm(aux);

And so on.

In short:

A=zeros(N);
k=sqrt(1/N);
A(1,:) = k*ones(1,N);
for i=2:N
    aux = [ones(1,i-1),-(i-1),zeros(1,N-i)];
    A(i,:) = aux/norm(aux);
end